Hundreds of thousands of out-of-work Americans are receiving their final unemployment checks sooner than they expected, even though Congress renewed extended benefits until the end of the year.

The checks are stopping for the people who have the most difficulty finding work: the long-term unemployed. More than five million people have been out of work for longer than half a year. Federal benefit extensions, which supplemented state funds for payments up to 99 weeks, were intended to tide over the unemployed until the job market improved.

In February, when the program was set to expire, Congress renewed it, but also phased in a reduction of the number of weeks of extended aid and effectively made it more difficult for states to qualify for the maximum aid. Since then, the jobless in 23 states have lost up to five months’ worth of benefits.

Next month, an additional 70,000 people will lose benefits earlier than they presumed, bringing the number of people cut off prematurely this year to close to half a million, according to the National Employment Law Project. That estimate does not include people who simply exhausted the weeks of benefits they were entitled to.

Separate from the Congressional action, some states are making it harder to qualify for the first few months of benefits, which are covered by taxes on employers. Florida, where the jobless rate is 8.7 percent, has cut the number of weeks it will pay and changed its application procedures, with more than half of all applicants now being denied.

The federal extension of jobless benefits has been a contentious issue in Washington. Republicans worry that it prolongs joblessness and say it has not kept the unemployment rate down, while Democrats argue that those out of work have few alternatives and that the checks are one of the most effective forms of stimulus, since most of it is spent immediately.

After the most recent compromise reached in February, another renewal seems unlikely.

The expiration of benefits is one factor contributing to what many economists refer to as a “fiscal cliff,” or a drag on the economy at the end of this year when tax cuts and recession-related spending measures will all come to an end unless Congress acts. The Congressional Budget Office warned last week that the combination could contribute to another recession next year.

Candace Falkner, 50, got her last unemployment check in mid-May, when extended benefits were curtailed in eight states. Since then she has applied for food stamps and begun a commission-only, door-to-door sales job. Since losing her job two years ago, Ms. Falkner said, she has earned a master’s degree in psychology and applied for work at numerous social service agencies as well as places like Walmart, but no offers came.

Ms. Falkner, who lives on the outskirts of Chicago, said she was grateful for the checks she received. But when they ended, she said, “They should have had some program in place to funnel those people back into the job market. Not to just leave them out there cold, saying, ‘The job market has improved, but there’s still 60,000 people in the city who can’t find one.’ ”

Unemployment is lower than it was when the emergency unemployment extensions were ramped up in November 2009. Now, it is 8.1 percent, down from 9.9 percent then. But it is still far higher than pre-recession norms, and there are more than three job seekers for every opening.

Proponents of extended benefits say the cuts are premature. Chad Stone, the chief economist at the liberal Center on Budget and Policy Priorities, said Congress had never before put the brakes on extended benefits when the labor market was so weak. “It’s moving in the wrong direction, and it’s occurring at a time when unemployment is very high,” he said.

Conservative economists and political leaders have argued that unemployment benefits prolong joblessness and simply transfer wealth from one area of the economy to another without contributing to growth.

Kevin A. Hassett, director of economic policy studies at the conservative American Enterprise Institute, said, “I haven’t liked the 99-week solution from the beginning because it creates an environment where people are subsidized to become a structural unemployment problem.”

Still, he is troubled by the latest developments. “If you just reduce the weeks of unemployment for people already unemployed but don’t do anything else, it’s a bad deal,” he said, “because they’re already about the worst-off people in society.”

He points to alternatives like using unemployment money to encourage entrepreneurship or paying benefits in a lump sum, rather than over time, to encourage people to find work faster.

Most states offer 26 weeks of unemployment benefits, plus the federal extensions that kicked in after the financial crash.

The number of extra weeks available by state is determined by several factors, including the state’s unemployment rate and whether it is higher than three years earlier. So states like California have had benefits cut even though the unemployment rate there is still almost 11 percent.

“Benefits have ended not because economic conditions have improved, but because they have not significantly deteriorated in the past three years,” Hannah Shaw, a researcher at the Center on Budget and Policy Priorities, wrote in a blog post. In May, an estimated 95,000 people lost benefits in California.

After the recession, 99 weeks became a symbol of the plight of the jobless, with those who exhausted their benefits calling themselves “99 weekers” or “99ers.” But by the end of September, the extended benefits will end in the last three states providing 99 weeks of assistance — Nevada, New Jersey and Rhode Island.

Some states have tightened eligibility as well. Nationwide, most people apply for benefits by phone. Last August, Florida began requiring people to apply online and to complete a 45-minute test to assess their job skills, according to a complaint submitted to the federal labor secretary by the National Employment Law Project and Florida Legal Services.

The complaint said that applicants with limited Internet access or English skills, disabilities or difficulty reading had effectively been shut out, and that failure to complete the assessment was illegally being used to deny benefits. Denials have soared; now just over half of applicants are rejected. Nationally, 30 percent of applicants are rejected, according to the law project.

The changes have saved the state $2.7 million, according to James Miller, a spokesman for the Florida Department of Economic Opportunity. The state’s unemployment rate, he pointed out, has declined for 10 straight months. “The Department of Economic Opportunity provides accommodations to individuals with barriers to filing their claims,” he wrote in an e-mail. “D.E.O. welcomes any review and is certain that Florida’s statutory changes are in full compliance with federal law.”

The Labor Department is reviewing Florida’s unemployment program in response to multiple complaints, a spokesman said.

Warning: First, this is a somewhat modified repost from — oddly enough — four years ago. Second, this post has math in it. A lot. Some of it might even be correct. If you are mathophobic, then you might want to skip to the end, where I reveal what Rosebud means.

And for those of you who are incredibly anal, yes, I know I kinda lost track of significant digits about 2/3 of the way through this. I was using a calculator, and just used whatever numbers it gave me to the last decimal place, leaving off for the most part trailing 0s. Sue me. I’m free on February 29th, 4800.

When I was a kid, I had a friend whose birthday was on February 29th. I used to rib him that he was only 3 years old, and he would visibly restrain himself from punching me. Evidently he heard that joke a lot.

Of course, he was really 12. But since February 29th is a leap day, it only comes once every four years.

And why is it only a quadrennial event?

Duh. Astronomy!

The Days of Our Lives

We have two basic units of time: the day and the year. Of all the everyday measurements we use, these are the only two based on concrete physical events: the time it takes for the Earth to spin once on its axis, and the time it takes to go around the Sun. Every other unit of time we use (second, hour, week, month) is rather arbitrary. They’re convenient, but not based on independent, non-arbitrary events.

It takes roughly 365 days for the Earth to orbit the Sun once. If it were exactly 365 days, we’d be all set! Our calendars would be the same every year, and there’d be no worries.

But that’s not the way things are. There are not an exactly even number of days in a year; there are about 365.25 days in a year. That means every year, our calendar is off by about a quarter of a day, an extra 6 or so hours just sitting there, left over. After four years, then, the yearly calendar is off by roughly one day:

4 years at 365 (calendar) days/year = 1460 days, but

4 years at 365.25 (physical) days/year = 1461 days.

So after four years the calendar is behind by a day. That means to balance it out again we add that day back in once every four years. February is the shortest month (due to some Caesarian shenanigans), so we stick the day there, call it February 29th, the Leap Day, and everyone is happy.

Integral to the plot

Except…

The year is not exactly 365.25 days long. Our official day is 86,400 seconds long. I won’t go into details on the length of the year itself (you can read a wee bit about it here), but the year we now use is called a Tropical Year and it is 365.242190419 days long. With malice aforethought — my calculator won’t hold that many digits — let’s round it to 365.2421904.

So it’s a bit short of 365.25. That hardly matters, right?

Actually, it does, over time. Even that little bit adds up. After four years, we don’t have 1461 physical days, we have

4 years at 365.2421904 (real) days/year = 1460.968762 days.

That means that when we add a whole day in every four years, we’re adding too much! We should really only add 0.968762 days. But that’s a bit of a pain, so we add in a whole day.

So even though we add a Leap Day in to balance the calendar, it’s still a bit off. It’s a lot better, for sure, but it’s still just a hair out of whack. This time, it’s ahead (since we added a whole day which is too much) by

1 – .968762 days = 0.031238 days, or about 45 minutes.

That’s not a big deal, but you can see that eventually we’ll run into trouble again. The calendar gains 45 minutes every 4 four years. After we’ve had 32 leap years (128 years of calendar time) we’ll be off by a day again!

So we need to adjust our calendar again. But 128 years is hard to remember, so it was decided to round that down to 100 years. After a century, we’ll have added that extra 45 minutes in 25 times (every four years for 100 years = 25 leap years). To be precise, after 100 years the calendar will be off by

25 x 0.031238 days = 0.780950 days.

That’s close enough to a whole day.

Confused yet? Here’s another way to think about. After 100 years, we’ll have had 25 leap years, and 75 non leap years. That’s a total of

But in reality we’ve had 100 years of 365.2421904 days, or 36524.2421904 days. So now we’re off by

36,525 – 36524.21904 = .78096

which, within roundoff error, is the number I got above. Woohoo.

So after 100 years, the calendar has gained almost a whole day on the physical number of days in a year. That means we have to stop the calendar and let the spin of the Earth catch up. To do this, every 100 years we don’t add in a leap day! To make it simpler, we only do this in years divisible by 100. So 1700, 1800, and 1900 were not leap years, we didn’t add an extra day, and the calendar edged that much closer to matching reality.

And so we’re good, right? Well…

But notice, he says chuckling evilly, that I didn’t mention the year 2000. Why not?

Because even this latest step isn’t quite enough. Remember, after 100 years, the calendar still isn’t off by a whole number. It’s ahead by 0.78095 days. So when we subtract a day by not having leap year every century, we’re overcompensating; we’re subtracting too much. We’re behind now, by

1 – 0.780950 days = 0.21905 days.

Arg! So every 100 years, the calendar lags behind by 0.21905 days. If you’re ahead of me here (and really, I can barely keep up with myself at this point), you might say “Hey! That number, if multiplied by 5, is very close to a whole day! So we should put the leap day back in every 500 years, and then the calendar will be very close to being right on the money!”

What can I say? My readers are very smart, and you’re exactly correct. So, of course, that’s not how we do things.

Instead, we add the leap day back in every 400 years! Why? Because if there is a stupid way to do something, that’s how it will be done.

After 400 years, we’ve messed up the calendar by 0.21905 days four times (once every 100 years for 400 years), and so after four centuries the calendar is behind by

4 x 0.21905 days = 0.8762 days

and that’s close enough to a whole day. So every 400 years February 29th magically appears on the calendar, and once again the calendar is marginally closer to being accurate.

Sanity check

As a check, let me do the math a second way, in the same method I did for the leap century gambit above. In 400 years we’ve had 303 non-leap years, and 97 leap years. The total number of days is therefore

We can see the remainder is 0.8762 days, which checks with the previous calculation, and so I’m confident I’ve done this right. (phew)

Of course, the calendar’s still not completely accurate at this point, because now we’re ahead again. We’ve added a day, when we should have added only 0.8762 days, so we’re ahead now by

1 – 0.8762 days = 0.1238 days.

Funny thing is, no one worries about that. There is no official rule for leap days with cycles bigger than 400 years. I think this is extremely ironic, because the amount we are off every 400 years is almost exactly 1/8th of a day! So after 3200 years, we’ve had 8 of those 400 year cycles, so we’re ahead by

8 x 0.1238 days = 0.9904 days.

If we then left leap day off the calendars again every 3200 years, we’d only be behind by 0.0096 days! That’s phenomenally accurate. I can’t believe we stopped at 400 years.

OK, so how does all work again?

But despite that, we’re done! We can now, finally, see how the Leap Year Rule works:

What to do to figure out if it’s a leap year or not:

We add a leap day every 4 years, except for every 100 years, except for every 400 years. In other words…

If the year is divisible by 4, then it’s a leap year, UNLESS

it’s also divisible by 100, then it’s not a leap year, UNLESS FURTHER

the year is divisible by 400, then it is a leap year.

So 1996 was a leap year (The Little Astronomer was almost born on leap day that year, in fact). 1997, 1998, and 1999 were not. 2000 was a leap year, because even though it is divisible by 100 it’s also divisible by 400.

1700, 1800, and 1900 were not leap years, but 2000 was. 2100 won’t be, nor 2200, nor 2300. But 2400 will be.

This whole 400-year thingy was started in the year 1582 by Pope Gregory XIII. That’s close enough to the year 1600 (which was a leap year!), so in my book, the year 4800 should not be a leap year.

But who listens to me? If you’ve gotten this far without blowing out your cerebrum, then I guessyou listen to me. All this is fun, in my opinion, and if you have gotten this far you know as much about leap years as I do.

Which is probably too much. All you really need to know is that this year is a leap year, and we’ll have plenty more for some time. You can go through my math and check me if you’d like…